### On subspaces of separable first countable ${T}_{2}$-spaces

G. Reed (1976)

Fundamenta Mathematicae

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G. Reed (1976)

Fundamenta Mathematicae

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G. Reed, P. Zenor (1976)

Fundamenta Mathematicae

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Roman Pol (1977)

Fundamenta Mathematicae

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Richard Hodel (1975)

Fundamenta Mathematicae

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Eric van Douwen, Teodor Przymusiński (1980)

Fundamenta Mathematicae

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Wojciech Olszewski (1990)

Fundamenta Mathematicae

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K. Alster (1981)

Fundamenta Mathematicae

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Alan Dow, Oleg Pavlov (2006)

Fundamenta Mathematicae

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Hušek defines a space X to have a small diagonal if each uncountable subset of X² disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω₁ which has a small diagonal will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable; in particular, Juhász and Szentmiklóssy...

Karol Pąk (2009)

Formalized Mathematics

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We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindelöf spaces and state the above theorem in this special case. We also introduce the...

Aleksander V. Arhangel&#039;skii, Raushan Z. Buzyakova (2002)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that if a regular space $X$ is the union of a finite family of metrizable subspaces then $X$ is a $D$-space in the sense of E. van Douwen. It follows that if a regular space $X$ of countable extent is the union of a finite collection of metrizable subspaces then $X$ is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a $D$-space. Some other new results on the properties of spaces which are unions of a finite collection of...